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Trisecant flops, their associated \(K3\) surfaces and the rationality of some cubic fourfolds. (English) Zbl 1521.14032

A very general cubic fourfold is conjectured to be irrational and the locus of rational ones, also conjecturally, should be the union of certain irreducible divisors \(\mathcal{C}_d\) (in their moduli \(\mathcal{C}\)), of special admissible cubic fourfolds of discriminant \(d\); their rationality relies on the existence of certain \(K3\) surface (further references on this Kuznetsov Conjecture can be found in the Introduction of the paper under review). In this paper, the authors take the point of view of Mori Theory to describe the cases \(d=14, 26, 38\) and \(42\), in fact the first four admissible values of \(d\) (cubics fourfolds in \(\mathcal{C}_d\), \(d=14,26,38\), were known to be rational); and furthermore to prove (see Theorem 5.12) the rationality of every cubic fourfold in \(\mathcal{C}_{42}\), the first not known case. The birational maps from \(X\) to a rational fourfold \(W\) are displayed in a, say, Mori Theory diagram (see (0.1)) in such a way that the role of the \(K3\) surface is very explicit: a non-minimal birational model in \(W\) can be constructed via some very peculiar linear systems of hyperplane sections.

MSC:

14E08 Rationality questions in algebraic geometry
14M20 Rational and unirational varieties
14M07 Low codimension problems in algebraic geometry
14N05 Projective techniques in algebraic geometry
14J28 \(K3\) surfaces and Enriques surfaces
14J70 Hypersurfaces and algebraic geometry
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